holds. The research on extending this result led Saharon Shelah to the invention of PCF theory. Galvin gave an elementary proof of the Baumgartner–Hajnal theorem (). The original proof by Baumgartner and Hajnal used forcing and absoluteness. Galvin and Shelah also proved the square bracket partition relations and . Galvin also proved the partition relation where η denotes the order type of the set of rational numbers.